数学代写|代数学代写Algebra代考|MATH355

数学代写|代数学代写Algebra代考|Directed Graphs and Multigraphs

It is often the case that the relationship between objects is not entirely symmetric, and when this happens it is useful to be able to distinguish which object is related to the other one. For example, if we try to use a graph to represent the connectivity of the World Wide Web (with vertices representing web pages and edges representing links between web pages), we find that the types of graphs that we considered earlier do not suffice. After all, it is entirely possible that Web Page A links to Web Page B, but not vice-versa-should there be an edge between the two vertices corresponding to those web pages?

To address situations like this, we use directed graphs, which consist of vertices and edges just like before, except the edges point from one vertex to another (whereas they just connected two vertices in our previous undirected setup). Some examples are displayed in Figure 1.29.

Fortunately, counting paths in directed graphs is barely any different from doing so in undirected graphs – we construct the adjacency matrix of the graph and then look at the entries of its powers. The only difference is that the adjacency matrix of a directed graph has a 1 in its $(i, j)$-entry if the graph has an edge going from vertex $i$ to vertex $j$, so it is no longer necessarily the case that $a_{i, j}=a_{j, i}$.

As an even further generalization of graphs, we can consider multigraphs, which are graphs that allow multiple edges between the same pair of vertices, and even allow edges that connect a vertex to itself. Multigraphs could be used to represent roads connecting cities, for example-after all, a pair of cities often has more than just one road connecting them.

Multigraphs can be either directed or undirected, as illustrated in Figure 1.30. In either case, the method of using the adjacency matrix to count paths between vertices still works. The only difference with multigraphs is that the adjacency matrix no longer consists entirely of $0 \mathrm{~s}$ and $1 \mathrm{~s}$, but rather its entries $a_{i, j}$ describe how many edges there are from vertex $i$ to vertex $j$.

数学代写|代数学代写Algebra代考|Systems of Linear Equations

Much of linear algebra revolves around solving and manipulating the simplest types of equations that exist-linear equations:

A linear equation in $n$ variables $x_1, x_2, \ldots, x_n$ is an equation that can be written in the form
$$a_1 x_1+a_2 x_2+\cdots+a_n x_n=b,$$
where $a_1, a_2, \ldots, a_n$ and $b$ are constants called the coefficients of the linear equation.
For example, the following equations are all linear:
Note that even though the top-right equation above is not quite in the form described by Definition 2.1.1, it can be rearranged so as to be in that form, so it is linear. In particular, it is equivalent to the equation $4 x-6 y=-3$, which is in the desired form. Also, the bottom-left and bottom-middle equations are indeed linear since the square root and trigonometric functions are only applied to the coefficients (not the variables) in the equations. By contrast, the following equations are all not linear:

In general, an equation is linear if each variable is only multiplied by a constant: variables cannot be multiplied by other variables, they cannot be raised to an exponent other than 1 , and they cannot have other functions applied to them.
Geometrically, we can think of linear equations as representing lines and planes (and higher-dimensional flat shapes that we cannot quite picture). For example, the general equation of a line is $a x+b y=c$, and the general equation of a plane is $a x+b y+c z=d$, both of which are linear equations (see Figure 2.1).

代数学代考

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数学代写|代数学代写Algebra代考|Systems of Linear equation

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$n$变量$x_1, x_2, \ldots, x_n$中的线性方程可以写成
$$a_1 x_1+a_2 x_2+\cdots+a_n x_n=b,$$
，其中$a_1, a_2, \ldots, a_n$和$b$是称为线性方程系数的常数。

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